Anna and Hannah have $\$80$ each. Their friend offered to invest their money, promising to return a sum $r$ times as great as what they invested. Anna was suspicious, so she invested $\$20$ only, but Hannah invested her entire $\$80$. Fortunately, the friend did indeed return a sum $r$ times as great to each. They decided to make another investment. This time, Hannah invested all of the money returned to her, and Anna invested the money returned to her and the remaining $\$60$. Again, they got a sum $r$ times as great as what they invested. In the end, Hannah had twice the amount Anna had. Write an equation in terms of $r$ that models the situation.
Explanation: The strategy We know that in the end, Hannah had twice the amount Anna had. If we let $H$ denote Hannah's final amount and $A$ denote Anna's final amount, we obtain the equation $H=2A$. Now, let's express $H$ and $A$ in terms of $r$. Expressing Hannah's final amount With the first investment, Hannah invested $\$80$ and received an amount $r$ times what was invested, or $80r$ dollars. Hannah then invested this amount, $80r$ dollars, and again received an amount $r$ times what was invested, or $80r\cdot r=80r^2$ dollars back. Expressing Anna's final amount Anna invested $\$20$ at first and received $20r$ dollars back. Then she invested this and the remaining $\$60$. So her total second investment was $20r+60$ dollars and so she received $(20r+60) r=20r^2+60r$ dollars back. Putting things together We found that $H=80r^2$ and $A=20r^2+60r$. Since $H=2A$, we can substitute and find an equation in terms of $r$ that models the situation. The answer is: $ 80r^2=2(20r^2+60r)$